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There appears to be a good deal of misinformation swimming around on this topic. The vector/scalar bosons, quarks, leptons and neutrinos if you consider them non-leptons are currently assumed to have mass from similar phenomena which are somehow tied to the Higgs mechanism. When people say that 'most of the mass is coming from somewhere else' they are only referring to compound particles (hadrons) such as protons and pions. The Higgs field and it's corresponding vacuum potential (VEV) are ad hoc additions to the standard model and their inclusion is not based on deeper principles or observations. The case for boson masses is easier to swallow since the idea is that the introduction of electroweak symmetry breaking would produce a scalar boson (e.g. Higgs) via Goldstone's theorem and generate non-zero mass eigenstates that mix with the W and Z bosons. The current idea is that since fermions can interact with the higgs and thus there exists some yukawa-like term (scalar-fermion interaction), there may also be a term that combines the higgs vacuum potential with fermions in a way that is similar to how mass terms are usually described in toy model QFTs. These terms contain coupling constants which simultaneously determine how strongly a fermion is coupled to the Higgs and how massive said fermion is. This explanation is good enough since without these terms there is no obvious way of including fermion masses in the standard model but the microscopic origin of these varying coupling and VEV values has not been settled
